Για πρωινό έφτιαξα ομελέτα με κρεμμύδια και λίγα καρότα.

Yes. Για means for and it commonly introduces a purpose/occasion phrase:

• Για πρωινό = for breakfast / as breakfast (i.e., “what I had/made for breakfast”). It’s a very natural way to frame what you ate or prepared for a meal.

Greek often omits the article in set phrases like this, especially after certain prepositions, when speaking generally:

• Για πρωινό = “for breakfast (in general)” If you add the article, it becomes more specific/emphatic:
• Για το πρωινό = “for the breakfast (we’re talking about)” / “for breakfast (this particular one)”

It’s accusative. Many Greek prepositions (including για) normally take the accusative:

• για + accusative → Για πρωινό
  So you’re not seeing a “direct object” here; it’s the case required by the preposition.

Έφτιαξα is aorist (simple past) of φτιάχνω (“to make/prepare”). The aorist presents the action as completed:

• έφτιαξα = “I made (and it was done)” If you used the imperfect (έφτιαχνα), it would suggest an ongoing/repeated action in the past (“I was making / I used to make”).

Greek is a pro-drop language: the subject pronoun is often omitted because the verb ending already shows the person/number.

• έφτιαξα = “I made” (1st person singular) You can add εγώ for emphasis or contrast:
• Για πρωινό, εγώ έφτιαξα… = “For breakfast, I made…”

Φτιάχνω is broad: “make, prepare, fix.” In food contexts it often means “make/prepare” and can imply cooking if relevant, but it’s not as specifically “cook” as μαγειρεύω.

• έφτιαξα ομελέτα is very natural for “I made an omelette.”

Greek can omit the article with foods/objects in some contexts, especially when introducing what you made/had, similar to “I made omelette” vs “I made an omelette.” You can absolutely say:

• έφτιαξα μια ομελέτα = “I made an omelette” (more explicitly “one”)
• έφτιαξα την ομελέτα = “I made the omelette” (a specific one already known)

Ομελέτα is feminine (η ομελέτα). So adjectives/determiners agreeing with it would be feminine:

• μια νόστιμη ομελέτα = “a tasty omelette”

Με means with and it takes the accusative:

• με κρεμμύδια
• με … καρότα This is the standard way to list ingredients/accompaniments.

Κρεμμύδια is neuter plural (“onions”). Greek often uses plurals for ingredients in a generic/quantity sense (“with onions”). Singular is possible but changes the nuance:

• με κρεμμύδι = “with onion” (often meaning “with some onion,” or focusing on it as an ingredient in general)

Λίγα agrees with καρότα, which is neuter plural:

• λίγα (neuter plural) + καρότα (neuter plural) Other forms would match different genders/numbers:
• λίγες
  • feminine plural noun
• λίγοι
  • masculine plural noun
• λίγο
  • neuter singular / “a little”

Yes. Και is “and,” used just like English to join words/phrases:

• με κρεμμύδια και λίγα καρότα = “with onions and a few carrots” You could also omit the second με in Greek (as here), and it still applies to both items.

Yes. Greek accents show the stressed syllable and are part of correct spelling:

• έφτιαξα stress on έ-
• κρεμμύδια stress on -μμύ-
• καρότα stress on -ρό- Wrong stress can sound unnatural and sometimes can create ambiguity with other words in Greek.